A remarkable periodic solution of the three-body problem in the case of equal masses

Abstract

Using a variational method, we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane. The orbit has zero angular momentum and a very rich symmetry pattern. Its most surprising feature is that the three bodies chase each other around a fixed eight-shaped curve. Setting aside collinear motions, the only other known motion along a fixed curve in the inertial plane is the “Lagrange relative equilibrium” in which the three bodies form a rigid equilaterial triangle which rotates at constant angular velocity within its circumscribing circle. Our orbit visits in turns every “Euler configuraiton” in which the three bodies form a rigid equilaterial triangle which rotates at constant angular velocity within its circumscribing circle. Our orbit visits in turns every “Euler configuration” in which one of the bodies sits at the midpoint of the segment defined by the other two (Figure 1). Numerical computations by Carles Simó, to be published elsewhere, indicate that the orbit is “stable” (i.e. completely elliptic with torsion). Moreover, they show that the moment of inertia $I(t)$ with respect to the center of mass and the potential $U(t)$ as functions of time are almost constant.